How many digits does the number 2^100 have? Using complicated caculator softwares it is 31 digits but how will you find out with pencil and paper

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- April 18th 2011, 04:11 PMsundance240How many digits?
How many digits does the number 2^100 have? Using complicated caculator softwares it is 31 digits but how will you find out with pencil and paper

- April 18th 2011, 06:18 PMSoroban
Hello, sundance240!

Quote:

How many digits does the number 2^100 have?

Using complicated caculator softwares, it has 31 digits,

but how will you find out with pencil and paper?

We note that: .2^{10} .= .1024 .≈ .10^3

Hence: .2^{100} .≈ .(10^3)^{10} .= .10^{30}

. . which has 31 digits.

- April 19th 2011, 01:09 AMOpalg
In prehistoric times we used to do problems like this by using logarithms to base 10. These were used so often that everybody knew that log(2) = 0.3010... . Therefore log(2^{100}) = 100*log(2) = 30.10..., and since this is between 30 and 31 it follows that 2^{100} is a 31-digit number.

- April 19th 2011, 02:06 AMemakarov
- April 19th 2011, 05:21 AMtopsquark
- April 19th 2011, 06:34 AMDeveno
- April 19th 2011, 12:51 PMHallsofIvy